7.9, due April 1

At first I really didn't like the new notation for permutations, but as I kept reading I eventually got used to it and it does seem simpler although I am still really glad that it isn't the notation we used from the beginning. I do not like/understand well the factorization of a permutation. I don't know if I just can't follow what order they are doing them in or what but for some reason when they write out the factorization I can't seem to follow what they've done or how the two sides are equal. Because of this, it was also hard for me to follow the last two proofs of this section.

7.8, due March 30

I'm a little confused since we are mixing multiple subsets with quotient groups. It's just hard for me to keep track of what group(s) various elements are part of as I go through the proofs. I do like the vocabulary for simple groups because the term seems the concept in my mind. It's pretty cool that there are only FIVE nonabelian simple groups of order less than 1000! The stuff on the last page is kind of confusing to me mostly because we again have so many subsets. Also, my favorite line from the section was, "Yes, Virginia, there is a Second Isomorphism Theorem" Haha! so random!

7.7, due March 28

Quotient groups? I had a hard enough time with quotient fields...Mostly I see the notation in the proofs and theorems and it just doesn't mean anything to me even though they defined it. I also feel like quotient groups only have notation in common with quotient fields which is very confusing to me. Also, mixing ideals into groups seems a little questionable. I suppose it's cool that we've defined quotient groups and I'm sure I'll see some interesting similarities with fields tomorrow (?).

7.6 (second half), due March 25

This section really isn't that bad, the trickiest part is probably just keeping straight when certain things hold for normal subgroups and thus remembering that sometimes they don't hold (multiplying congruence classes). I think it's cool when nonabelian subgroups end up being normal because you can't tell that they will be right off the bat. Also, the equivalence statements are pretty easy to use, and are not even THAT difficult to prove.

7.6 (through page 211), due March 23

I can't tell if I don't understand this section because it's especially difficult or if it's just because I'm so exhausted from taking the test. I do know, however, that I don't like that we have now have left congruence classes. (though at least they both have the same properties) I feel like I'll definitely be getting these mixed up and especially mix the interpretations/definitions for each. So the only good thing that I see is kind of what I said before that at least they hold the same properties of reflectivity, symmetry, transitivity and disjoint/identical property. hoorah.

Test 2 questions, due March 21

I think the most important theorems are the named ones and the other one that we are expected to be able to state and prove. It'll also be important to know all the axioms for rings and groups-how to prove something is a group. I expect the types of questions to be similar to that of the first test: one or two proofs and then computational (although I'm not entirely sure what computational problems will look like for these sections). One thing I am only starting to grasp is the idea of cosets, I can't visualize them at all which is really putting a block on my understanding. Any problem/extra explanation regarding section 9.4 would be helpful for me since that section didn't really fit with the rest of what we did around that time.

7.5 (second half), due March 18

All of the statements we can make about isomorphisms based on the order of the group is interesting (and probably useful). I think the reason I find it interesting is that I wasn't expecting them to work out like that. These isomorphisms also make it a little easier to visualize groups and wrap my head around them a little better. None of this section was overly difficult, I'm still stuck on the first half of 7.5. I didn't fully understand it I suppose and the homework was incredibly difficult for me. From the second half reading the hardest part would probably be the second half of the proof of theorem 7.3 just because it's so long and a little tricky to follow since it sort of considers lots of possibilities.

7.5 (first half), due March 16

Congruence classes of groups is really difficult for me to wrap my mind around, I know that with some practice it'll make sense, but I'm not there yet. I think just seeing/working with some different examples would clear this up for me. It's also somewhat confusing to have right cosets (and so do we have left cosets?). It is nice that we have a lot of the same terms/vocabulary just applied in new ways but that they generally work the same with some minor adjustments.

Make up blog (Dr. Vitaly Bergelson)

Honestly, most of Dr. Bergelson's lecture went straight over my head. He began by talking about a set with finite volume which was a concept I am unfamiliar with. He then went on to discuss theories that involved the densities of sets which was still something I don't understand although I quickly saw that the even numbers have a density of 1/2 and there is a pattern that follows (although I don't know what this concept means). Part of the proof he was showing involved the statement that the density of all the prime numbers is zero which broke the pattern. Anyway, he was talking about combinatorics, so I didn't necessarily expect to understand most of what he was saying. He mainly focused on Poincare's recurrence theorem, which I at least found the history of as somewhat interesting, but generally speaking I think my math vocabulary and concept understanding just wasn't high enough to really understand where he was going.

7.4, due March 14

I'm really glad the definition for isomorphism concerning groups really didn't change. And, theorem 7.18 is so cool! Even after reading the proof, I don't COMPLETELY understand why it's true but that makes it that much cooler of a theorem. I think a lot of the theorems in this section will be really useful and they're interesting because I didn't expect them to hold. What's confusing for me so far is permutations vs representations and the advantages of each.

7.3, due March 11

The proof of theorem 7.11 is one of the hardest things for me in this section because I don't have a good understanding of analyzing the order of an element using modulo or a product of 2 other numbers. I read over those theorems from the last section multiple times and they just don't make that much sense to me. I'm starting to get it, but I don't think I know it well enough to think to use it for a proof yet. I like the cyclic groups because it's an easy way to quickly come up with a subgroup and the proof of why it is a subgroup makes perfect sense! (and it's convenient that all of these are abelian) It's also really great that we've finally made the connection between the order of an element and the order of a group, it's a pretty cool connection too!

7.2, due March 9

What's great about this section is that it basically tells us that we groups generally work the way we want/expect them to. Most of the theorems that we would expect to hold true are addressed in this section and are, in fact, true. Oh joy! It's a little weird that we use the term "order" to describe the number of elements in a group and in the way it's used in this section (describing a single element) although this isn't particularly difficult, just different. Basically I think I understand this section and it won't be too bad!

7.1 (second half), due on March 7

It's cool that we can define groups using geometry transformations (that isn't something I would have thought of).  It's also cool that a ring with identity ALWAYS has at least one subset that is a group under multiplication. The hardest part to wrap my mind around  was the last example because they defined the cartesian product of real numbers and D4. Since I can't visualize this, it was hard for me to think about as a group.

7.1 (through page 164), due on March 4

The only thing I really don't understand is what a symmetric group is. The book introduced it by example right at the end of a paragraph and really didn't go into any other detail about what exactly it means. Otherwise this section pretty much made sense. The idea of a group is a little hard to grasp just because we haven't seen that many examples and the definition seems some what vague since a group is formed with just some operation.

6.3, due March 2

I like this section because it seems fairly intuitive after already considering Rings modulo primes (or irreducible polynomials). I also really like the way of thinking about primes with ideals because it makes sense! It's also cool that using this definition some surprising things come out to be prime-such as polynomials with even constant terms. I don't understand as much why you can sometimes have a quotient ring modulo a prime that is not a field, when previously that was always a field. I understand the counter example and that we can still say it is an integral domain, it just seems counter-intuitive for it to not necessarily be a field.